Multi-objective adjoint optimization for drag reduction of very low Earth orbit satellites

To optimize aerodynamic shapes of very low Earth orbit (VLEO) satellites operating in rarefied atmospheric environments, a gas-kinetic adjoint optimization method was developed, and multi-objective adjoint optimization research was conducted on the three-dimensional shape of the satellite body. The optimization targeted the main body of mini-satellites with dimensions comparable to "GOCE" and "SLATS", operating at altitudes of 150—300 km. In the optimization, the generatrix of the axisymmetric body was parameterized using the CST method, with the objectives of reducing aerodynamic drag and increasing volume, while considering multiple geometric constraints on satellite length, maximum diameter, and slenderness ratio. The rarefied gas flow around the satellite was described by the BGK model equation combined with the diffuse reflection boundary condition. The corresponding adjoint equations were constructed to enable efficient computation of design sensitivities, and the set of optimal shapes on the Pareto front was finally obtained via a gradient descent algorithm. Optimization results show that the developed adjoint optimization method converges within about 10 optimization steps (20 solutions of kinetic equations), and the solution for a single Pareto point takes approximately 42 minutes. As the volume increases, the optimal satellites mainly exhibit two classes of shapes— "water-drop" shapes and cylindrical shapes with blunt leading and trailing edges—with a distinct shape transition region in between, where the final solution is determined by the knee point method. Compared to a cylindrical baseline of equal volume and constraints, the optimized shapes achieve drag reductions of approximately 11.0%～22.6%. Moreover, the optimal shapes at altitudes of 150 km and 300 km show little difference. This study provides valuable experience for the aerodynamic shape design of VLEO satellites, validating the high efficiency, effectiveness, and application potential of adjoint optimization in rarefied flow design problems.